Problems & Puzzles: Puzzles

Puzzle 1130 As 68 Long time ago I published the following Curio "The largest known number that is the sum of two distinct primes in exactly two ways: 68 = 7 + 61 = 31 + 37." Q. Can you get the largest known number that is the sum of k>1 distinct prime in exactly k ways?

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During the week from 6 to 12 of May, 2023, contributions came from Alessandro Casini, JM rebert, Emmanuel Vantieghem, Oscar Volpatti

 

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Alessandro wrote:

K=2: 68 is confirmed

K=3: 130 = 2 + 19 + 109 = 2 + 31 + 97 = 2 + 61 +67

K=4: 42 = 3 + 5 + 11 + 23 = 3 + 7 + 13 + 19 = 5 + 7 + 11 + 19 = 5 + 7 + 13 + 17

K=5: 59 = 3 + 5 + 7 + 13 + 31 = 3 + 5 + 11 + 17 + 23 = 3 + 7 + 13 + 17 + 19 = 5 + 7 + 11 + 13 + 23 = 5 + 7 + 11 + 17 + 19

K=6: 76 = 3 + 5 + 7 + 11 + 13 + 37 = 3 + 5 + 7 + 11 + 19 + 31 = 3 + 5 + 7 + 13 + 17 + 31 = 3 + 5 + 7 + 13 + 19 + 29 = 3 + 7 + 11 + 13 + 19 + 23 = 5 + 7 + 11 + 13 + 17 + 23

K=7: no number found

Later he added:

 expanded the survey. So, I report all the solutions that i’ve found for K=2 - 16.

 

K=2: 16 - 18 - 20 - 22 - 26 - 28 - 32 - 62 - 68

K=3: 26 - 27 - 29 - 32 - 36 - 42 - 46 - 48 - 54 - 58 - 60 - 100 - 124 - 130

K=4: 33 - 35 - 38 - 40 - 42

K=5: 55 - 59

K=6: 59 - 63 - 74 - 76

K=7: no solution found <15000

K=8: no solution found <15000

K=9: 124 - 153 - 161

K=10: 159 - 192

K=11: 233

K=12: 227

K=13: 276

K=14: no solution found <5000

K=15: 372 - 425

K=16: 480

Let me point out that some number is solution for two different K: 26 (K=2-3), 32 (K=2-3), 42 (K=3-4), 59 (K=5-6), and 124 (K=3-9). Now I wonder if K=7-8-14 has any solution.

 

For K=16, 480

 

3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 101

3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 71 + 83

3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 53 + 59 + 89

3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 61 + 67 + 73

3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 47 + 53 + 61 + 83

3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 47 + 53 + 71 + 73

3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 47 + 59 + 67 + 71

3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 53 + 59 + 61 + 71

3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 43 + 47 + 53 + 59 + 83

3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 41 + 43 + 47 + 53 + 59 + 79

3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 41 + 43 + 47 + 53 + 67 + 71

3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 41 + 43 + 47 + 59 + 61 + 71

3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 37 + 41 + 43 + 47 + 53 + 59 + 73

3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 37 + 41 + 43 + 47 + 53 + 61 + 71

3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 71

3 + 5 + 7 + 11 + 13 + 17 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61

 

 

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Jean-Marc wrote:

I found the following numbers for 1 < k <=6.

68 = 7 + 61 = 31 + 37

 

130 = 2 + 19 + 109 = 2 + 31 + 97 = 2 + 61 + 67

 

42 = 3 + 5 + 11 + 23 = 3 + 7 + 13 + 19 = 5 + 7 + 11 + 19 = 5 + 7 + 13 + 17

 

59 = 3 + 5 + 7 + 13 + 31 = 3 + 5 + 11 + 17 + 23 = 3 + 7 + 13 + 17 + 19 = 5 + 7 + 11 + 13 + 23 = 5 + 7 + 11 + 17 + 19

 

76 = 3 + 5 + 7 + 11 + 13 + 37 = 3 + 5 + 7 + 11 + 19 + 31 = 3 + 5 + 7 + 13 + 17 + 31 = 3 + 5 + 7 + 13 + 19 + 29 = 3 + 7 + 11 + 13 + 19 + 23 = 5 + 7 + 11 + 13 + 17 + 23

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Emmanuel wrote:

In the next table you find for some values  k  the biggest  m  I could find
such that  m  is a sum of  k  primes in exactly  k  ways (but there might be bigger solutions) :

k        m            
3      130
4      42
5      59
6      76
9      161
10    192
11    233
12    227
13    276
15    372

For  k = 7, 8  and  14  I found no such  m.

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Oscar wrote:

I searched up to k = 10.


k = 3, n = 130
130 = 2 + 19 + 109
130 = 2 + 31 + 97
130 = 2 + 61 + 67


k = 4, n = 42
42 = 3 + 5 + 11 + 23
42 = 3 + 7 + 13 + 19
42 = 5 + 7 + 11 + 19
42 = 5 + 7 + 13 + 17


k = 5, n = 59
59 = 3 + 5 + 7 + 13 + 31
59 = 3 + 5 + 11 + 17 + 23
59 = 3 + 7 + 13 + 17 + 19
59 = 5 + 7 + 11 + 13 + 23
59 = 5 + 7 + 11 + 17 + 19

k = 6, n = 76
76 = 3 + 5 + 7 + 11 + 13 + 37
76 = 3 + 5 + 7 + 11 + 19 + 31
76 = 3 + 5 + 7 + 13 + 17 + 31
76 = 3 + 5 + 7 + 13 + 19 + 29
76 = 3 + 7 + 11 + 13 + 19 + 23
76 = 5 + 7 + 11 + 13 + 17 + 23

k = 7, no n in exactly 7 ways

k = 8, no n in exactly 8 ways

k = 9, n = 161
161 = 3 + 5 + 7 + 11 + 13 + 17 + 23 + 29 + 53
161 = 3 + 5 + 7 + 11 + 13 + 19 + 23 + 37 + 43
161 = 3 + 5 + 7 + 11 + 13 + 19 + 29 + 31 + 43
161 = 3 + 5 + 7 + 11 + 17 + 19 + 23 + 29 + 47
161 = 3 + 5 + 7 + 13 + 17 + 19 + 23 + 31 + 43
161 = 3 + 5 + 7 + 13 + 17 + 19 + 29 + 31 + 37
161 = 3 + 5 + 11 + 13 + 17 + 19 + 23 + 29 + 41
161 = 3 + 7 + 11 + 13 + 17 + 19 + 23 + 31 + 37
161 = 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 37

k = 10, n = 192
192 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 41 + 53
192 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 29 + 41 + 47
192 = 3 + 5 + 7 + 11 + 13 + 17 + 23 + 29 + 31 + 53
192 = 3 + 5 + 7 + 11 + 13 + 17 + 23 + 29 + 37 + 47
192 = 3 + 5 + 7 + 11 + 13 + 17 + 23 + 29 + 41 + 43
192 = 3 + 5 + 7 + 11 + 13 + 19 + 23 + 31 + 37 + 43
192 = 3 + 5 + 7 + 11 + 17 + 19 + 23 + 29 + 31 + 47
192 = 3 + 5 + 7 + 11 + 17 + 19 + 23 + 29 + 37 + 41
192 = 3 + 5 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 41
192 = 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37

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Giorgos Kalogeropoulos wrote on June 8, 2023:
 

If we use the data from A344989 that asks for the smallest number with the same properties, we can conclude that

k=17: no solution

k=18: no solution

k=19: no solution

k=20: 752

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 67 + 71 + 73 + 79 + 83,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 59 + 61 + 71 + 73 + 79 + 83,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 53 + 59 + 61 + 67 + 71 + 79 + 83,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 83,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 61 + 67 + 73 + 83 + 89,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 61 + 71 + 73 + 79 + 89,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 59 + 61 + 67 + 71 + 79 + 89,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 53 + 59 + 61 + 67 + 71 + 73 + 89,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 89 + 97,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 73 + 83 + 97,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 67 + 71 + 79 + 97,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 73 + 79 + 101,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 61 + 67 + 71 + 73 + 101,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 83 + 103,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 71 + 79 + 103,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 67 + 71 + 73 + 103,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 79 + 107,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 71 + 73 + 109,

752 = 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 73 + 113
 

 

 

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